Quantum restricted Boltzmann machine is universal for quantum computation

Abstract

The challenge posed by the many-body problem in quantum physics originates from the difficulty of describing the nontrivial correlations encoded in the many-body wave functions with high complexity. Quantum neural network provides a powerful tool to represent the large-scale wave function, which has aroused widespread concerns in the quantum superiority era. A significant open problem is what exactly the representational power boundary of the single-layer quantum neural network is. In this paper, we design a 2-local Hamiltonian and then give a kind of Quantum Restricted Boltzmann Machine (QRBM, i.e. single-layer quantum neural network) based on it. The proposed QRBM has the following two salient features. (1) It is proved universal for implementing quantum computation tasks. (2) It can be efficiently implemented on the Noisy Intermediate-Scale Quantum (NISQ) devices. We successfully utilize the proposed QRBM to compute the wave functions for the notable cases of physical interest including the ground state as well as the Gibbs state (thermal state) of molecules on the superconducting quantum chip. The experimental results illustrate the proposed QRBM can compute the above wave functions with an acceptable error.

Provable Advantage in Quantum Phase Learning via Quantum Kernel Alphatron

“…Firstly, we design a learning instance with the same format as in Lemma 1. Since the quantum adiabatic algorithm with 2-local Hamiltonians can implement universal quantum computational tasks [32,64]…”

“…For the second question, we consider the solution of QPL using quantum computers. Among different applications of quantum computing [22,23,24,25,26,27,28,29,30,31,32], a variety of quantum machine learning algorithms [20,33,34,35,36,37,38,39,40,41,42] have been developed for different problems and demonstrated the potential for solving classically intractable problems, using parameterized circuits and classical optimization of noisy-intermediate-scale-quantum devices. Using a quantum computer, we propose the quantum kernel Alphatron algorithm to efficiently solve the QPL problem.…”